Question

for exercises 10-14, answer the questions. 10. how many unique triangles are there with angle measures 36°, 64°, and 80°? explain.

Explanation:

Step1: Check angle sum property

First, we check if the sum of the given angles is \(180^\circ\). The sum of the angles is \(36^\circ + 64^\circ+ 80^\circ\). Calculating that: \(36 + 64 + 80 = 180\). So the angles satisfy the triangle angle - sum property.

Step2: Analyze triangle uniqueness (AAA case)

When we know the three angle measures of a triangle (the AAA - Angle - Angle - Angle case), we are dealing with similar triangles. However, for the number of unique triangles (up to congruence), if we only fix the angles, we can have infinitely many triangles with these angle measures because the sides can be scaled (dilated) to different lengths while still maintaining the same angle measures. But if we consider triangles up to similarity, there is exactly one unique triangle (in terms of shape) with these angle measures. But in the context of triangle construction (SSS, SAS, ASA, AAS criteria for congruence), when we have three angles (AAA), we can have infinitely many non - congruent triangles (since the side lengths can vary) but one unique triangle in terms of its angle - determined shape (similarity class). But usually, when the question is about "unique triangles" in the sense of non - congruent triangles, since we can have different side lengths (as long as the angles are fixed), we can form infinitely many triangles by varying the side lengths (scaling the triangle). But if we consider triangles with the given angle measures (regardless of side length, just the angle - determined triangle), the set of all triangles with these angles are similar, so there is one unique triangle (up to similarity) with these angle measures. But let's re - examine the angle sum: \(36+64 = 100\), \(100 + 80=180\), so the angles are valid for a triangle. In the case of AAA (three angles known), we can have infinitely many triangles that are similar (same shape, different sizes) because we can choose any positive length for one side and then determine the other sides using the Law of Sines (\(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k\), where \(k>0\) is a scale factor). So, if we consider triangles up to congruence (same size and shape), there are infinitely many non - congruent triangles (since \(k\) can be any positive real number). But if we consider triangles up to similarity (same shape), there is exactly one unique triangle (the similarity class) with these angle measures. However, in the context of a typical middle - school or high - school geometry question, when asked about the number of unique triangles with given angle measures (AAA), the answer is that there are infinitely many triangles because the side lengths can be any positive lengths (as long as the angle measures are fixed), and each different set of side lengths (different scale factor \(k\)) gives a non - congruent triangle. But if we consider the "shape" (similarity), there is one unique shape. But let's go back to the basic triangle construction: The Angle - Sum Property of a triangle states that the sum of interior angles is \(180^\circ\). Here, \(36 + 64+80 = 180\), so a triangle with these angles is possible. Now, for the number of unique triangles: When we have three angles (AAA), we can create triangles of different sizes (by changing the length of the sides) while keeping the angles the same. For example, we can have a very small triangle with sides of length \(a,b,c\) and a very large triangle with sides \(ka,kb,kc\) (\(k > 0\)) that both have angles \(36^\circ\), \(64^\circ\), and \(80^\circ\). So, in terms of non - congruent triangles, there are infinitely many. But if we consider triangles up to similarity (same shape), there is exactly one unique triangle (the similarity class) with these angle measures. But in most introductory geometry contexts, when the question is about "how many unique triangles" with given angle measures, and since the angles only determine the shape (similarity) and not the size, we can say that there are infinitely many triangles (because size can vary) or one unique triangle (up to similarity). But let's check the angle sum again to confirm the triangle is valid. \(36+64 = 100\), \(100 + 80=180\), so the angles are valid. Now, the key concept here is that for a triangle, if we know all three angles (AAA), we can have infinitely many triangles that are similar (same angles, different side lengths). So the number of unique triangles (in terms of non - congruent triangles) is infinite because we can scale the triangle (change the side lengths) while keeping the angles constant.

Answer:

There are infinitely many unique (non - congruent) triangles with angle measures \(36^\circ\), \(64^\circ\), and \(80^\circ\) because the three angles satisfy the triangle angle - sum property (\(36^\circ+64^\circ + 80^\circ=180^\circ\)) and we can create triangles of different sizes (by varying the side lengths) while maintaining these angle measures (using the Law of Sines, where the sides are proportional to the sines of their opposite angles, and we can scale the sides by any positive real number). If considering triangles up to similarity (same shape), there is one unique triangle with these angle measures.